where the entries in positions \((1,1)\) , \((2,1)\)  and \((i,i+1)\)  for all \(i\in \lbrace 1,\dots ,n\rbrace \)  are \(x_2\) 's and the other ones are \(x_1\) 's. Up to permutation of left columns and change of variables, \(\mathsf {Edge}_2\)  is the matrix \(\mathsf {Mal}\) , and thus \(\mathsf {Edge}_2 \Leftrightarrow _{\mathsf {lex}} \mathsf {Mal}\) . For a general \(n\geqslant 2\) , a variety has \(\mathsf {Edge}_n\) -closed relations if and only if its theory admits an \(n\) -edge term in the sense of [1]}. Therefore, as it is shown in [1]}, one has \(\mathsf {Cube}_n\Leftrightarrow _{\mathsf {alg}}\mathsf {Edge}_n\) . Using Proposition 1.7 of [3]}, we know that \(\mathsf {Edge}_n \Rightarrow _{\mathsf {lex}} \mathsf {Cube}_n\) . However, for a general \(n\) , the converse is not true. For instance, from the computer-based results of [4]}, we can see that \(\mathsf {Cube}_3\nRightarrow _{\mathsf {lex}}\mathsf {Edge}_3\) . This is an additional example of the context dependency of the implications between matrix properties.

